Warning: Possible readers who can appreciate this post are engineering and math majors who need calculator techniques for the board exams. Numbers and equations are involved. 😛
A few days ago, I got an email from a civil engineering student in Pangasinan. He asked me this question:
“I’ve read in your website that you are sharing tips for engineering board exam.. I’m so grateful to see people who share their knowledge and experiences to other people. I’m very interested to know some calculator techniques in solving problems in engineering mathematics like calculus using fx-991 es.. can you share some to me? thank you so much!”
Thanks so much for this emailed question! Keep the questions coming, board exam related or not, because I really accommodate what you guys want to read here next.
I already made a post on calculators as part of my series of board exam tips in this blog. But the techniques were not posted because I was too lazy at the time. LOL. I am not really much of an expert in maximizing this calculator model, but there were some tricks that I found useful.
One of the best advantages of using Casio ES 991 is the Natural Display. It looks exactly as one would write the equation on paper:
I strongly advise you to read the Casio ES 991 manual before you experiment with the techniques. I know, it’s like I am asking you to drink acid with this tip, but trust me, it works! For one, I learned from that long manual that you cannot use the derivative and integral function button in certain equations.
So many calculator users may rely too much on the results produced by the integral and derivative buttons of this calculator. But you know what? The algorithm used on the ES 991 calculator for derivatives and integrals is heavily based on a fair mathematical approximation (according to the manual!), and bound to fail in certain types of equations. Be careful. When I took the board exam last year, the only difference between me and the top 1 of our board exam was roughly around five questions. Every item counts.
Also, be consistent with the angle usage. Check if the mode of your calculator is properly set to degrees or radians. I heard of a tale of one board exam taker who was computing in degrees but her calculator was set to radians the whole time. She panicked during the exam because her answers were not found in the multiple choices given. She got to fix this minor error about 30 minutes before time was up. She still passed, because she is very intelligent, but that is truly a source of additional stress, if you ask me.
I already said something about the SHIFT + SOLVE function in the previous post on calculators, so I will not repeat them anymore.
I frequently used the CALC button, especially when I am forced to repeat an equation over and over again for different values. This is very common in long engineering problems.
For example, you need to find y for 5 values of x, and the equation is y = x + 5. I am using basic equations here just so you get the principle. For example, the given values for x are 1,15,25,30, and 16.
Manually, you will have to type in “1+5”, “15+5”, “25+5”, “30+5”, and “16+5” separately to get the 5 y values you need. That’s just too mechanical, and a complete waste of board exam time.
Using CALC, you can just use any of the variables in the calculator (A,B,C,D,X and Y are all usable for this purpose). Just type in “X+5” and press the CALC button and the equal sign. You will be prompted by the calculator to give the X values without having to type the formula over and over again. And since you can all use A, B, C, D, X, and Y, you can get values with as much as 6 variables without having to hurt your fingers.
Just be careful not to store important constant values in A, B, C, D, X, and Y when you are using it for CALC. Chances are, the values you stored will be deleted as you use the CALC button.
There are 8 modes in Casio ES 991.
Mode 1 COMPUTATION
This mode is the default mode for calculations and this is where most of your computations will occur.
Mode 2 COMPLEX
The complex mode is hardly used in my major, geodetic engineering. But we did have need to convert Polar to Rectangular coordinates in computing for lot data. I got this technique from Engr. Machele Felicen, who was our Top 1.
We just enter A<B + CALC
It will ask for the values of A and B (A is the r, B is the theta).
Then the answer it will produce is something like this:
1.45365444 + 3.4567i
The first term and second term are the rectangular coordinates (x,y) already.
That saves you helluva lot of time than when you use the one given by the Casio manual for converting (r, theta) into (x,y).
Mode 3 STAT
I did not get to use this much during my exam because we geodetic engineers are more inclined to geometry and calculus. But based on what I have seen so far, you can do so many things for Statistics with ES 991. And the natural display will easily show you the list of the values. It was unlike my old calculator where you have to scroll down a lot and punch a lot of buttons before you get to each n value.
Mode 4 BASE-N
BASE-N…I did not get to use it at all. LOL. Sorry.
Mode 5 EQUATION
Mode 5 was one of my favorites during the board exam. It allows you to get the values for 2 equations, 2 unknowns and 3 equations, 3 unknowns. It also will help you get the quadratic and cubic roots of an equation. They are most useful. All you have left to do during the exam is to reduce all the complex equations into any of these four forms and then you are super done with it. 😉
During our review class, there was one challenging math problem with 9 equations and three unknowns. That’s hell if you do it manually. Simplification was vital. I just reduced it to three equations and used Mode 5 and I got X, Y, and Z in less than 5 minutes. 😀
Mode 6 MATRIX
Matrix operations were pretty common, too. Too bad it can only handle 3 by 3 matrices at the most. But you can store up to three matrices and work with them without having to type the matrices over and over again.
Mode 7 TABLE
In my initial example for CALC, I used different X values which are far from each other in the number line. But what if the x values are within an interval and is equally spaced? For example, the X values are 1,2,3,4, and 5. The Table mode is ideal for values like these.
This TABLE mode will instantly prompt you to give an expression for f(X).
It will ask you where the X values will start (“Start?”) and end (“End?”). It will also ask the interval between the values (“Step?”). In the example’s case, Start is 1, End is 5 and Step is 1 unit.
For its output, the calculator will yield a table containing all the values that you need, showing both X and F(X) as you would see it tabulated on paper, thanks to natural display.
(Note: I tried using variables A, B, C, D, and Y for Table mode, but I believe it only works when you use the X variable.)
I also use the Table Mode when I have a hard time visualizing a certain function. I just choose a reasonable interval of values, plug the equation and I sketch the table of values on a scratch paper so that I have a better vision of the equation’s physical properties.
Mode 8 VECTOR
If you need dot products and dim, you can use this mode. It can hold three-dimensional data (X,Y,Z), even. I have not used it too much. I think it will be more useful for Physics classes than in Math board exams. 🙂 I leave the physicists to enlighten you on the matter.
My reader explicitly asked about Calculus techniques. For this, I believe ES 991 poses you with a tiny limitation. It will only get you derivatives and integrals for those with absolute numerical values. This means, you ought to have upper and lower limit values for the integration function. In college, this function may be a bit useless because you will need to solve calculus problems by hand if you really want to master the theorems and get a good grade in your class. Don’t do too much shortcuts in college at the expense of learning the principles. But during board exams, where there are application-focused problems likes Maxima-Minima and Related Rates, you can definitely take advantage of the derivative and integral buttons. 😉
Do use the calculator everyday, even for small things like your shopping list. It allows you to get used to the calculator buttons. This, in turn, improves your manual dexterity and speed in using the calculator. Make sure you have enough batteries. At the time of my exam, I replaced the batteries 2-3 days before the actual exam and I had a spare calculator of the same model in case my original calculator breaks down somewhere during the board exam.
It takes a lot of practice to learn shortcuts. And it saves time in getting you from point A to B during an exam.
But do make sure that you know the long methods before you play around with the shortcuts. I am totally not encouraging that you rely purely on shortcuts to pass the exam. The complete comprehension of the concepts is very, very important.
If you have other calculator tips for our dear readers, this post is open for discussion and comments.
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